Finite Group Databasebeta

← #13140522

Subgroups — #13140522

Order 1728.

493 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721850C211yesmaximal, characteristic
2#11721095C211yesmaximal, characteristic, complemented
3#63659⋊C3C212yesmaximal, complemented
4#11720324C212yesmaximal, complemented
5#11720303C211yesmaximal, characteristic, complemented
6#10591477—32nomaximal
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C2×#6235—22no—
9C2×C4⋊#272C2211yescharacteristic
10C4×#1084C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#88448—22no—
14C3×#940—32no—
15C3×#940—61no—
16#63869—32no—
17#63659D312yescomplemented
18C3×#832—32no—
19#63563—32no—
20#63562—32no—
21#63554—32no—
22C23×H(3,𝔽3)—21no—
23C22×#272—21no—
24D4×H(3,𝔽3)—22no—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26#6235—44no—
27C2×#1084D412yes—
28C4⋊#272—22no—
29C2×(C3⋊Dic3)⋊C3—22no—
30(C3⋊(C3⋊C8))⋊C3—22no—
31#5071—91no—
32#4764—92no—
33C2×C32×D4—21no—
34C2×C32×D4C2×D312yes—
35C2×C3×#67—62no—
36C2×C3×#67—62no—
37C2×C3×#67—64no—
38C2×C3×D12—32no—
39C2×C3×D12—61no—
40C4⋊(C3⋊Dic3)C2×D312yes—
41C2×C3⋊(C3⋊C8)C2×D312yes—
42C3×C4⋊Dic3—32no—
43C3×C4×Dic3—32no—
44C3×C4×Dic3—61no—
45C2×C3×C3⋊C8—32no—
46#3724—62no—
47Dic3⋊Dic3—62no—
48C22×H(3,𝔽3)—41no—
49C22×H(3,𝔽3)—42no—
50C22×H(3,𝔽3)C2×D411yescharacteristic
51C2×#272—41no—
52C2×#272—42no—
53C4×H(3,𝔽3)#4112yes—
54#1084—22no—
55(C3⋊Dic3)⋊C3—42no—
56C3×#127—91no—
57#940—32no—
58#940—61no—
59#890—92no—
60#870—91no—
61C3×#102—92no—
62#832—32no—
63#832—92no—
64#832—92no—
65#821—92no—
66#815—92no—
67#809—92no—
68#803—91no—
69C23×C32—22no—
70C23×C32—22no—
71C22×C3×D3—62no—
72C22×C3×D3—62no—
73C32×D4—22no—
74C32×D4—22no—
75C32×D4—42no—
76C2×C32×C4—21no—
77C2×C32×C4C22×D312yes—
78C2×C3⋊Dic3—22no—
79C3×#67—128no—
80C3×#67—124no—
81C3×#67—124no—
82C2×C3×Dic3—32no—
83C2×C3×Dic3—32no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—62no—
86C3×D12—62no—
87C3×D12—62no—
88C3×D12—122no—
89C3⋊(C3⋊C8)—22no—
90C3×C3⋊C8—62no—
91#490—271noSylow
92C2×H(3,𝔽3)—81no—
93C2×H(3,𝔽3)#12411yescharacteristic
94C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
95#272—81no—
96C2×C3×D4—32no—
97C2×C3×D4—61no—
98C2×C3×D4—91no—
99C2×C3×D4—182no—
100C2×C3×D4D3211yescharacteristic
101C2×#67—62no—
102C2×#67—62no—
103C2×#67—64no—
104C2×D12—32no—
105C2×D12—61no—
106C2×C3×C8—92no—
107C3×C4⋊C4—92no—
108C3×C42—91no—
109C4⋊Dic3—32no—
110C4⋊Dic3—92no—
111C4⋊Dic3—182no—
112Dic3⋊C4—182no—
113Dic3⋊C4—182no—
114C4×Dic3—32no—
115C4×Dic3—61no—
116C2×C3⋊C8—32no—
117C2×C3⋊C8—92no—
118C22×C32—21no—
119C22×C32—44no—
120C22×C32—42no—
121C22×C32—42no—
122C22×C32—42no—
123C22×C32—42no—
124C22×C32D3×D412yes—
125C2×C3×D3—122no—
126C2×C3×D3—124no—
127C2×C3×D3—122no—
128C2×C3×D3—122no—
129C2×C3×D3—122no—
130C32×C4—22no—
131C32×C4#23512yes—
132C32×C4#23712yes—
133C3⋊Dic3—42no—
134C3×Dic3—64no—
135C3×Dic3—62no—
136C3×Dic3—62no—
137C3×Dic3—122no—
138#127—91no—
139#125—271no—
140#102—92no—
141#102—272no—
142#97—271no—
143H(3,𝔽3)#49011yescharacteristic, complemented, Sylow
144C23×C3—21no—
145C23×C3—62no—
146C23×C3—62no—
147C23×C3—181no—
148C22×D3—62no—
149C22×D3—62no—
150C3×D4—22no—
151C3×D4—62no—
152C3×D4—62no—
153C3×D4—122no—
154C3×D4—182no—
155C3×D4—364no—
156C2×C3×C4—32no—
157C2×C3×C4—61no—
158C2×C3×C4—92no—
159C2×C3×C4—182no—
160C2×C3×C4C2×D3211yescharacteristic
161#67—124no—
162#67—124no—
163#67—128no—
164C2×Dic3—32no—
165C2×Dic3—32no—
166C2×Dic3—62no—
167C2×Dic3—62no—
168C2×Dic3—182no—
169D12—62no—
170D12—62no—
171D12—122no—
172C3×C8—182no—
173C3⋊C8—62no—
174C3⋊C8—182no—
175C2×C32—22no—
176C2×C32—21no—
177C2×C32—82no—
178C2×C32—82no—
179C2×C32#88512yes—
180C2×C32#91112yes—
181C2×C32#91412yes—
182C3×D3—242no—
183C3×D3—242no—
184C2×D4—91no—
185C2×D4—182no—
186C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
187C2×C8—92no—
188C4⋊C4—92no—
189C4⋊C4—542no—
190C42—91no—
191C22×C3—32no—
192C22×C3—42no—
193C22×C3—41no—
194C22×C3—61no—
195C22×C3—122no—
196C22×C3—122no—
197C22×C3—122no—
198C22×C3—124no—
199C22×C3—122no—
200C22×C3—362no—
201C22×C3—361no—
202C22×C3#381111yescharacteristic
203C2×D3—122no—
204C2×D3—122no—
205C2×D3—122no—
206C2×D3—122no—
207C2×D3—124no—
208C3×C4—32no—
209C3×C4—32no—
210C3×C4—62no—
211C3×C4—182no—
212C3×C4—362no—
213C3×C4#380412yes—
214Dic3—62no—
215Dic3—62no—
216Dic3—64no—
217Dic3—122no—
218Dic3—362no—
219C32—21no—
220C32#476412yescomplemented
221C23—21no—
222C23—181no—
223D4—22no—
224D4—182no—
225D4—364no—
226C2×C4—92no—
227C2×C4—182no—
228C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
229C8—182no—
230C2×C3—32no—
231C2×C3—32no—
232C2×C3—32no—
233C2×C3—61no—
234C2×C3—62no—
235C2×C3—81no—
236C2×C3—242no—
237C2×C3—242no—
238C2×C3—721no—
239C2×C3#6393712yes—
240C2×C3#6395311yescharacteristic
241D3—242no—
242D3—242no—
243C22—41no—
244C22—42no—
245C22—361no—
246C22—362no—
247C22#8867711yescharacteristic
248C4—182no—
249C4—362no—
250C4#8867112yes—
251C3—32no—
252C3—61no—
253C3#1059148011yescharacteristic
254C2—81no—
255C2—721no—
256C2#1172119312yes—
257C2#1172129411yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721850C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#490
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11721095C2semidirect
3. #63659⋊C3C2semidirect
4. #11720324C2semidirect
5. #11720303C2semidirect
17. #63659D3semidirect
143. H(3,𝔽3)#490semidirect
220. C32#4764semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117218502
#117210952
#63659⋊C32
#117203242
#117203032
#105914773